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Problem Statement-
Create 1---4 expressions for each of the numbers 1-5. If you find more than one answer for a number write that down too. In each case, the expression must use the numbers 1, , , & 4 exactly once.
Process-
First I just tried simple addition to figure out the problem.
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1+++4= 10
1+4=55
41+=74
Then I realized my numbers were too big, so I started to add parenthesis and multiplication to my problem.
++4x1=
(++1)4=4
4x+1+=15
4xx1+=14
4x+1+=1
4xx+1=5
(4x1)+(x)=10
(4x1+)x=18
Then I started adding subtraction to my problem.
4 -+-1=
4 -1x+=11
4 -1x+=5
4 -++1=4
-1+x4=8
4 -+-1=
4 --1+=4
41-=
-14=
4-1=
This gave me most of my answers after I played around a bit.
Solution-
1
4
5
6
7
8
10
11
1
1
14
15
16
17
18
1
0
1
4
5
Extensions-
To change the problem I could use different numbers instead of 1, , , &4 for instance 4, 5, 6, & 7.Then I could also change the range of numbers that I have to find solutions for like 10-5.
Evaluation-
I learned that there are lots of different ways to get just one solution and there are a lot of solutions for using just a few numbers in a math equation. It was tough, but I enjoyed working on it. I probably wouldn't change the problem like I said it was a challenge but I enjoyed working on it.
1.) Problem statement
How many eggs did the farmer start out with using the clues she gave her agent? Is there more than one possibility for the answer?
On the way to market a farmer hit a pothole and her cart spilled leaving every egg, that was once inside, broken.
To file a full report the farmer had to tell her agent all she could remember about how many eggs were in the cart. She couldn't remember the exact number of eggs so she told her agent what she knew about putting them into equal groups and what worked out best.
When she put them into groups of , , 4, 5, and 6 there was always one egg left over.
When she put them into groups of 7 they came out evenly.
.) Process
When starting to solve the problem I realized that the number had to be divisible by 7 but, couldn't be divisible by , , 4, 5, or 6. That meant that the number was going to be larger than 4. Then I started picking numbers randomly to try them out.
I then realized that for five to go into a number and have a remainder of one, the number had to end in an 1 or a 6. Well, then if the number ended in a 6 it would make it even and an even number wouldn't work because any even number is divisible by . All of this helped me conclude that my answer had to end in an 1. Then I just started randomly picking numbers that ended in 1 to see if they were divisible by 7.
Then if it was, I went on to see if the other numbers went into it leaving a remainder of 1.
.) Solution
Once I guessed around enough, I found that the farmer started out with 01 eggs. I know that my answer is right because it fits all of the information the farmer gave her agent. 01 is divisible by 7 and when it is divided by , , 4, 5, and 6 it always has a remainder of one. It helped to actually sit and think about it before I just started guessing numbers. It really saved me a lot of time when I figured out it had to end in an 1. There are also more possibilities but they would be higher numbers in the thousands at least.
4.) Extensions
Okay, say the farmer is still at home and she is trying to put her eggs into groups. She doesn't know how many eggs she has but she knows that her eggs will go into groups of seven evenly but when they are put into groups of , , 4, 5,and 6 it always comes out to the farmer having one egg left over. How many eggs does the farmer have? Or what if, instead of going evenly into groups of 7 it went in evenly into groups of eleven or another prime number?
5.) Evaluation
This problem seemed impossible in the beginning the first time I read over it. When I finally sat down and thought about it, I understood it better and ideas just started pouring into my head. Once I wrote something down onto the paper it lead me to new ideas which lead me to even more ideas and so on. I thought this problem was fun because I always enjoy a challenge. I don't think I would change the problem if I had the chance just because I wouldn't want it to be any easier. I learned that if you read over the problem and really think about it the answer might come to you over time. When I was thinking about the problem, after I had finished it, instead of going through all of work of dividing everything out I could have just used the basic rules of division to save time.
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